Thanks For Not Stacking Me, Bro! (Part 6)

OK, granted this guy almost stacked me, but I just found this hand representative of general donkey play, where they play draws loose/passively out of position, and then fail to get full value when their draw hits.

Seat 2: UTG+2 ($152.55)
Seat 5: bruechips (CO) ($289.20)
Seat 6: Button ($303.25)
Seat 7: SB ($223.20)
*** HOLE CARDS ***
Dealt to bruechips [Ah As]
UTG+2 calls $2
CO raises to $10
Button calls $10
SB calls $9
UTG+2 folds (This was a pretty juicy table. I wasn't crazy about my seat because the button was a LAG who, while I think is a losing player, could put me in some tough spots. The SB was a complete donkey, playing very loose. This UTG+2 idiot was limping in early position and then folding to a raise in a 4-way pot where he's closing out the action. Donkerrific. In this hand I'm obviously feeling good with aces, but I'm also going to have to be aware that I could lose a big pot.)

*** FLOP *** [8d 9h 4d]
SB bets $22
CO raises to $58 (I would be SHOCKED if the SB were leading out here for 2/3 pot with a set. Shocked. I would even be surprised if he did it with top two pair. I think he has some one-pair hand like J9 or K8 or a draw like every time. I'm way ahead of his range and I don't want to call and give the button good odds to call with QT or something....although the button is aggro enough to try a squeeze play with a pretty wide range if I just called, and I'd have a sweet opportunity to come over the top. In any case, I decided to charge the draws and second best hands by raising for value.)
Button folds
SB calls $36

*** TURN *** [8d 9h 4d] [3c]
SB checks
Button bets $92 (Beautiful turn card and he checks. I'm pretty sure he's drawing to somewhere between 5 and 12 outs. There is about $145 in the pot and he has $155 left in his stack. I could just shove, but I thought he'd be more likely to call, or maybe even shove over top of, a smaller bet. If he were guaranteed to double up when he hits, then if I bet an amount x, he calls that amount x to win 145 (the amount already in the pot) + x (my turn bet) + 155-x (what's left in his stack after calling the turn bet). So the implied odds would be (145 + x + 155 - x):x = 290:x. So if he has 12 outs (for instance if he has QdTd) for 12/44 = 3/11 equity (i.e., an 8:3 dog to catch up on the river), he makes a mistake by calling anytime 290/x is less than 8/3, or x > 109. If he has 8 outs for 8/44 = 2/11 equity, he should not call if 290/x is less than 9/2, or x > 65. And remember these calculations assume that he gets all-in on the river anytime he catches up on the river, and never puts in another chip if he misses. So by betting $92 I do give him implied odds with his biggest draws, but I force him to make a big mistake with any lesser draws, and if he ever shoves in over the top of a $92 bet where he would have folded to a shove, that's just gravy. Also note that there are a lot more 8 and 9-out draws than 12-out draws. Any JT or 76 gives 8 outs, any two diamonds (including A-high diamonds, since I do NOT have the Ad) except for QT, 75, and QJ of diamonds makes 9 outs, whereas only the three diamond hands mentioned there give 12 outs. If he had more than 12 outs either on the flop or turn, for instance JT of diamonds, 67 of diamonds, 9x of diamonds, I think he'd just shove either the flop or the turn, in which case my bet sizing doesn't matter, as long as I bet. If he would be calling a shove very often then obviously shoving is better, but he timed down for a while before calling this bet, which makes me think he probably would have folded to a shove. If I had it to do over I might bet just a touch more, like $99, but I think anywhere in the $90-100 range is good, as it is tempting for him to call, while still being a bad mistake, and might give him the illusion of some fold equity.)
SB calls $92
*** RIVER *** [8d 9h 4d 3c] [Qs]
SB bets $18
bruechips calls $18 (ROR!!! He has $63 left in his stack, the pot is now $330ish, and he bets $18. I considered going ahead and putting him in as he could be putting in a blocking bet with Qx of diamonds, but I decided to take the safer route and just call to avoid value-owning myself vs. JT.)

*** SHOW DOWN ***
SB shows [Th Jh] a straight, Queen high

OK, so he won this hand by getting the straight on the river, but he played it absolutely atrociously. His bet/call on the flop is terrible. His check/call on the turn is terrible (the math I did up there to show the biggest bet he can call, remember, assumes that he will get full value on the river), and his small river value bet is even worse. I guess he's trying to induce me to shove for another $45? I'm pretty much NEVER shoving that river to try to get a calling station to fold for another $45 in a $350 pot. If I ever shove over the top of his bet, it would be for value, in which case I'm calling if he shoves anyway (I would definitely have called if he had shoved the river). His inability to get value on the river makes his turn call even worse. Well played donkey!!!

-BRUECHIPS

Art of the Min Raise (Part 14)

Usually this series is about ridiculing donkeys in cash games that either min check-raise sets on very drawy boards or min 3-bet AA and KK preflop. But I do occasionally use the min-raise myself, particularly in tourneys. You want to make bets smaller in general in tourneys because you have to be more conservative with your chips, as you can't buy any more. Whereas in a cash game 100 bbs deep usually the min-raise means a monster (or sometimes air on a dry board), in tourneys (and sometimes vs. shorties in a cash game), the min raise often means a weak made hand such as top pair-weak kicker or even A-high. For me personally, I most often use it when I call a pre-flop raise in position, usually in a blind battle, and want to min-raise a c-bet. Here are a couple of examples:

Seat 5: SB (1,455)
Seat 6: bruechips (BB) (1,500)

*** HOLE CARDS ***
Dealt to bruechips [Ad Jd]
7 folds...
SB raises to 90
BB calls 60 (I'm very tight early in tourneys so I don't want to re-raise here although I would often in a cash game, but AJs way too strong to fold to a SB open.)

*** FLOP *** [Kc 9d 2s]
SB bets 90
bruechips raises to 180 (This is a good spot to min-raise because it's a dry board that he'll be c-betting every time, and this is how I would play weak-ish made hands such as KJ as well as 99 and 22. I'm also just forcing him into a really tough spot. Even if he has a hand as strong as JJ, he's faced with the possibility of either re-raising and getting shoved on by all the hands that beat him and folding out all the hands that are behind, or calling and having potentially to face a couple of big turn and river bets. By putting in 180, I force him to make a decision for a much larger portion of his stack.)
SB folds

Just to prove that I don't have just bluffs there:

Seat 4: SB (5,420)
Seat 5: bruechips (BB) (5,345)

*** HOLE CARDS ***
Dealt to bruechips [Td Kd]
SB raises to 600
BB calls 300 (Deep enough to just call and use position here.)

*** FLOP *** [Tc Ks 6h]
SB bets 900 (c-bet he'd make every tiem)
bruechips raises to 1,800 (Now I'm hoping he has KQ, or that he has nothing and the min-raise just makes him go nuts, since he knows that I know he'd c-bet every time, and thinks I have AJ or J9 or something.)
SB raises to 2,700 (Sure enough he min 3-bets like a complete donkey)
bruechips raises to 4,720, and is all in (He looked committed enough to me, so I shoved...calling the 3-bet and just getting it all in on the turn is fine too, but just calling the 3-bet with 2,000 behind might set off alarm bells for him, whereas shoving might look bluffy)
SB calls 2,020
SB shows [8s 8d] (Sure enough, the min raise made this idiot go nuts)
*** TURN *** [Tc Ks 6h] [4c]
*** RIVER *** [Tc Ks 6h 4c] [8c]

Unfortunately, this donkey nails a two-outter on the river to bounce me from this tourney. Fawking donkeys....

-BRUECHIPS

Book Review: 'Winning Poker Tournaments' by Rizen, Apestyles, and Pearljammer

As part of my foray into tournament play, I picked up the new tourney book by online pros Eric "Rizen" Lynch, Jon "Apestyles" Van Fleet, and Jon "Pearljammer" Turner. You can use points to get it from the Full Tilt Store. It's volume 1 of a planned 2-volume series. This volume covers hands up to the money bubble, with the next volume projected to cover hands from the money bubble to the end of the tournament. The book has 4 sections. In the first three, each author takes 50-60 hands and analyzes them action by action. In the final section, there are 20 hands where each author gives their own independent take on each action.

Of the three, Pearljammer's play is probably the most closely directed towards tournaments. By that I mean that his play is often more directed towards survival and conservatism rather than aggression and applying maximum pressure on opponents. For instance, he flat calls a pre-flop min raise with TT on the button, essentially playing the hand for set value rather than wanting to create a big pot early, even though he probably has the best hand. He even open limps TT from the hijack (which, of course, looks like a great play in the hand shown because he flops a boat). Later on in the group section, he decides to check his option in the BB with ATo after it's folded around to the SB, who limps in. He does manage to get in some thin river value bets, for instance value-betting Q5o on the river into two players on a Q63hhJ2 board. He obviously has a stellar record as a tourney player, so far be it from me to criticize his play, but he's a little passive pre-flop for my tastes.

Apestyles has a game that's probably most similar to the tourney game I am striving for. For instance, we find Apestyles raising TT UTG (granted, with blinds a bit higher than in the PearlJammer hand), whereas PearlJammer was limping it in the hijack. Whereas PearlJammer checks behind ATo in the BB, Apestyles mulls over raising in the same spot with Q4o. He eventually decides to check behind, and gets some very nice value raising the river on a K96KQ board. His section ends with a series of hands where he goes through every hand around the bubble of a big tourney as he really abuses the table to chip up, squeezing with QTo and 4-betting K7o in a blind battle. He also goes through some chip equity calculations in some hands, which none of the other authors do. I guess most readers wouldn't be interested in such calculations, which is why they are eschewed by PearlJammer and Rizen, but for a serious player, they're really crucial.

Rizen's section has some nice examples of 3-barrel bluffs (he fires 3 barrels at a KT5ddKT board), and some re-steals (T6o vs. the cutoff). The T6o is interesting because he 3-bets the cutoff, the BB shoves, and Rizen folds despite getting 2.4:1. His reasoning is that the BB's range is very very strong so that he might not even have the 29% equity he'd need to make the call chip-neutral, and also calling and losing would take away his chip lead at the table, whereas he could fold and maintain his chip lead. This is one of the only spots where the authors hint at the difference between chip EV and payout-EV. I guess the image effect of calling and having to show T6o and folding getting 2.4:1 is about the same, although he doesn't mention it.

Overall, I found many of the hands helpful, and I'll probably adjust my tourney play to be a little less risky pre-flop on the advice of PearlJammer. I don't think I'll ever find myself open-limping TT in late position, but I'll probably limp behind 88 or maybe even TT and AJs with a couple of limpers in front early on in a tourney, whereas in cash games I will be raising them pretty much every time. If you're a tourney player with some experience and a decent handle on ABC poker, this book will probably help you a lot, both to hone the ABC TAG style and to add in a few tricks. If you're a more advanced analytical player, you'll almost certainly find enough interesting material to make the book worth reading. I was interested enough that I will probably get the second volume. However you won't find any great material on calculating payout EV in terms of your stack, other stacks, and your assessment of your relative skill, which is really the central issue in tournament poker that separates it from cash games. As far as I know, Sklansky's 'Tournament Poker for Advanced Players' remains the most important book on this subject.

-BRUECHIPS

Brack is Beautiful (Part 11)

Got back into the full ring action tonight which produced this hand....I'm still not completely sure about the way I played the hand....please comment with any suggestions. I give my reasoning for playing this way, but I am willing to be convinced that other lines are better. The main villain in the hand seemed fairly solid with TAGish stats. We had tangled twice before, once where he had 3b my open, I 4b and he folded, and another time where he cold 4b my 3b of another player and I folded.

Seat 3: bruechips (CO) ($208.70)
Seat 5: SB ($206.25)
Seat 6: BB ($239)
Seat 7: UTG ($74.70)

*** HOLE CARDS ***
Dealt to bruechips [8s 9s]
UTG calls $2
bruechips calls $2 (This seems kind of ghey. Maybe my donk-n-go experiences are starting to infect my cash game pray. But the reason I limped behind instead of raising was that UTG is short, and I had seem him limp-re-raise once before. If he were full-stacked, I'd definitely raise to isolate him, but with stacks this shallow and him limping UTG, I decided to just limp behind. The button was not a very aggressive player who would be raising a lot, and the blinds are full-stacked, so I'm happy to take a flop with them. In general, my range for limping right here is suited connectors, smaller pocket pairs, and the occasional suited ace.)
SB calls $1
BB checks

*** FLOP *** [Js 7s 4d]
SB checks
BB checks
UTG checks
bruechips bets $6 (This is pretty standard, everybody checks to me and I bet my straight flush draw.)
SB folds
BB calls $6 (I'm pretty sure this does not represent a monster hand, as big hands would probably raise on this drawy a flop. I'm thinking he has something like a weak jack, maybe an 8d7d for middle pair and some backdoor equity, A7 is possible, 56, and spade draws, although I think most of the time he'd be playing draws more aggressively.)
UTG folds

*** TURN *** [Js 7s 4d] [Ac]
BB checks
bruechips bets $14 (Some would argue for checking behind here to be sure you get to see the river with your monster draw. But I decided it was worth betting here to try and take him off a weak one-pair hand, or get him to just call with that hand and let me win a bigger pot on the river, either by firing a third barrel or making the best hand and getting value. Also my draw is not to the nuts (unless the Ts hits), and I want to maintain the betting lead against bigger draws, again either to fold them out now or fire again on the river. This is certainly what I would do with 77, 44, A7s, A4s, and AJs, which are all in my range, although AJs probably less so as I'd usually raise it pre-flop.)
BB raises to $28 (This is a very odd min raise. I almost think misclicks make up a significant portion of his range. What really strong hand would he slowplay on this flop and then give such great drawing odds on the turn? This feels more like a kind of informational raise to see if his J9 or whatever is good. Or maybe he's taking some very weird line with QsTs or 2s3s for a flush draw that picked up a straight draw. The only way I could see the ace improving his hand is if he has A7, or if he has an A-high spade draw. The A-high spade draw would be a particularly ugly scenario for me. But again, I think it's highly unlikely that he'd play a big draw like the nut flush draw so weakly on the flop when he's up against somebody who limped preflop and bet when checked to. I'm definitely not folding to this raise, the question is whether I should call or raise. I decided to raise for a few reasons. First, there are big hands that I can very easily represent here, mainly sets and aces up. If he flopped bottom two and decided to take a cautious line of waiting for a safe turn card before raising. Is he really excited about getting it in here? Second, he probably expects me to fire a second barrel at the ace with my entire range, so there's a good chance that he's trying to make a cheap rebluff with a pretty weak hand. Third, if I just call and miss, I can't represent much on the river. I think he's a pretty good player who is not going to just check/fold a blank river. He will probably be firing another barrel, and I can't represent much if I raise to try and rebluff. If I had a big hand already on the turn, I'd raise it there. If I had a mediocre hand like AhTh and decided I didn't want to fold, I'd just call the turn and river instead of raising. If I do hit a spade, it's pretty obvious what I have if I bet or raise. Getting value will be difficult unless he has a better flush, in which case he'll be the one getting value from me. The only really great card would be a T. That would give me a well disguised hand that I could get paid off with.)
bruechips raises to $96 (As far as the sizing, I want to raise an amount that appears to commit me to the pot, so he won't feel like he has any fold equity at all with KsQs or something like that. I want those hands to fold, or even better, call and check/fold the river. At the same time, I think shoving or raising too much looks kind of weak and he might just get really curious and decide QJ is gold. If I had a set, I'd raise to about this amount to charge the draws and force him to make a decision for his whole stack, so that's the amount I choose here.)
BB has requested TIME
BB folds

He tanked for a while. He could have been just Hollywooding, not wanting to let me and the rest of the table know that he just tried to pull one over on me with complete air. Or he could have made some huge laydown. It's such a weird line for him to take. I'm still kind of wondering what he had there. My best guess is still that he had a weak made hand, and he wanted to try and end the hand there rather than check/call or check/fold the river. Any other ideas?

-BRUECHIPS

P.S. - One other interesting tid-bit, just to show you how many draws there are on this board, EVERY unpaired suited connector or one-gapper has a straight draw by the turn.

The Shortstack Game (Part 4)

First of all....I hate Blogger. I had a lengthier post written that was for some reason deleted by Blogger when I tried to change the font color. Thanks Blogger.

Anyway, it seems like nobody's really interested in this Shortstack Game. But having come this far, I should give some sketch of a solution for equilibrium in the game. If you are enterprising and want to find it (or at least one of them):

1) Show that there is no equilibrium when player 2 shoves less than 1/3 of the time

2) Show that there is no equilibrium when player 2 shoves more than 1/3 of the time

3) Find an equilibrium where player 2 shoves exactly 1/3 of the time.

In the equilibrium I found, player 1 raise/folds 27% of the time, raise/calls 29% of the time, and open folds 44% of the time. Player 1's expected payoff in equilibrium is .42, while player 2's is 1.08, showing the disadvantage of having to act first.

I can also solve for an n-player game where two of the players have to post blinds, and the other players sequentially must raise or fold if no-one has yet raised, shove or fold if a player has raised already, and fold or call if a player has shoved already.

If you are interested, post a comment or email spritpot.

-BRUECHIPS

THIN! (Part 11)

Well, since the Shortstack Game seems to getting very little interest, I figured I'd put in a THIN! post before going back to it.

And this hand actually does have something to do with the Shortstack Game, just because it's blind vs. blind in a tourney where both players are fairly short. As loyal readers know, late in tourneys I think you really need to maximize your fold equity and steal chips preflop as often as you can.

One of the best times is when it's folded to the small blind, who limps into your big blind. Donks love to limp into the small blind with any two cards, and often min bet any flop or some donkey move like that. In general I do not permit this and will be either shoving preflop or shoving the flop if I catch any piece at all.

Some donks try to combat this by limping their big hands like AA, KK, AK, etc., which I think is probably a pretty good strategy. However these hands don't come around nearly often enough to deter me from shoving over their limps a large portion of the time. Perhaps some of them realize this and start limp/calling with some more marginal hands, like A6o. I'm pretty sure this is a terrible strategy. Say stacks are 10 bbs and you're comparing limp/calling A6o in the small blind to just shoving in your 10 bbs. A6o has only 51% equity vs. the top 50% of all hands, which is probably about my shoving range:

equity win tie pots won pots tied
Hand 0: 51.043% 47.73% 03.32% 5805749604 403268034.00 { A6o }
Hand 1: 48.957% 45.64% 03.32% 5551921944 403268034.00 { 33+, A2s+, K2s+, Q2s+, J4s+, T6s+, 96s+, 86s+, 76s, 65s, A2o+, K5o+, Q7o+, J7o+, T8o+, 98o }

Whereas my calling range if the sb shoves is obviously much smaller, probably only 8% of all hands. Since you guys are probably bored of equity calculations, I'll spare you, but I don't think limp/calling a marginal hand like A6o is a good play.

An even worse play, which I can't come up with any logic for but which seems to be an important part of a tourney donk's arsenal, is limp/calling a terrible hand. I mean I really just don't get it. I guess they just get frustrated that I shove every time they limp like a donkey? I dunno. But something led to this "accidental value bet":

Seat 3: Donk (SB) (4,645)
Seat 4: bruechips (BB) (4,580)

*** HOLE CARDS ***
Dealt to bruechips [Kc 5s]
Donk calls 200 (does he really expect to get away with this BS?)
bruechips raises to 4,580, and is all in (gimme the pot, donk!)
Donk calls 4,180 (whoops...looks like I'm gonna have to spike something this time...)
Donk shows [Jc 5c] (actually...huh?!?!?)

*** FLOP *** [9h Ts Ah]
*** TURN *** [9h Ts Ah] [Qd]
*** RIVER *** [9h Ts Ah Qd] [Jh]
bruechips wins the pot (9,160) with a straight, Ace high

Just to clarify, J5s is a slight dog to a RANDOM hand. ROR!!! Oh man...donks make me laugh...

-BRUECHIPS

The Shortstack Game (Part 3)

Just in case anybody is still reading this...having set up our game and solved for player 2's best response function, we now need to figure out player 1's best reponse function. I ended the last post with a couple of questions to get you started considering player 1's best response function:



What's player 1's best response if player 2 folds every time?



Pretty obvious that if player 2 always folds, player 1 should always raise.



What's player 1's best response if player 2 shoves every time?



It's pretty obvious that if player 2 shoves every time, player 1 should never raise/fold. So depending on his hand, he should either raise/call or fold. If he folds, he gets zero, whereas if he raise/calls, he gets 21.5e - 20*(1-e). Set this expression equal to zero and solve for e, you get 20/41.5. So he should raise/call anytime his hand has more than (20/41.5) equity against a random hand. We already determined in the last post that these hands are 22+,A2+,K2+,Q3o+,Q2s+,J7o+,J3s+,T8o+,T6s+, and 97s+, for 54.8% of all hands. Player 1's average payoff is then:



.452*0 + .548*(.572*21.5 - .428*20) = 2.05



Whereas player 2 has an average payoff of -0.55. Not coincidentally, this is the exact inverse of the payoffs when player 1 raise/calls every time and player 2 best responds.



What if player 2 shoves 22+,A9o+,A5s+,KQo,KJs+ (14.6% of all hands) and folds everything else?



We will have to solve this by "backward induction", considering the last decision first and then using that to figure out the payoffs involved in making earlier decisions. So let's assume player 1 has raised and player 2 has shoved. Which hands should player 1 be calling with? He would be calling 17 to win 24.5, so he needs 24.5*e - 17*(1-e) > 0, or e greater than 40.9%. These hands are: 33+, ATo+, A9s+,KQs (9.7% of all hands). All other hands, had player 1 raised them, would be folded. Player 1's calling range then has 53.8% equity against player 2's shoving range. Now we can consider player 1's decision to raise or fold as his first move. Say he has the most marginal hand with exactly 40.9% equity. Should he raise? His expected payoff to raising is:



.854*1.5 + .146*(.409*21.5 - .591*20) = .84



This is clearly better than 0, the payoff from folding. Also note something else. If player 1 has a hand that he will NOT call a raise with, his payoff to raising is:



.854*1.5 + .146*(-3) = .84



This is also better than folding. That the payoff to raise/calling and raise/folding with the 40.9% equity hand is no coincidence. The 40.9% mark is exactly the point where calling player 2's shove and folding to it are equal in expected value.



Player 1's expected payoff to this strategy is now:



.854*1.5 + .146*(.097*(21.5*.538 - 20*.462) - .903*(1.5)) = 1.1



What is player 1's best response to any given strategy for player 2?



Let f be the % of the time that player 2 folds when player 1 raises. Then player 1's expected payoff of raise/calling is:



1.5*f +(1-f)(-3) = 4.5f - 3



This obviously equals 0 when f equals 2/3. So if player 2 is folding two-thirds of the time or greater, player 1 should raise every time. If player 2 shoves, then player 1 should call if he has greater than 40.9% equity vs. player 2's shoving range.



If player 2 folds less than 2/3 of the time (and therefore shoves more than 1/3 of the time), player 1 should never raise/fold, and should raise/call if:



1.5*f + (1-f)*(21.5e - 20*(1-e)) > 0



Where e is the equity of player 1's hand against player 2's shoving range.



-BRUECHIPS

The Shortstack Game (Part 2)

WARNING: This post is long and has a lot of math in it. If you don't understand something or just want to discuss, I encourage you to post in comments. I will try to respond and clarify.

In an earlier post I described a game representing the interaction between a cutoff raiser, player 1, and a shortstack on the button, player 2. Towards the end of the post, I asked two questions - what is player 2's best response if player 1 plays RAISE/FOLD every hand? What if he plays RAISE/CALL every hand?

If player 1 plays RAISE/FOLD every hand, it's pretty clear that player 2 should be shoving every hand. That is, SHOVE is player 1's best response to RAISE/FOLD. It doesn't matter what player 2's cards are, since player 1 will fold every time.


What if player 1 always plays RAISE/CALL? Player 2 has two options, FOLD and SHOVE. FOLD gives a payoff of zero, whereas SHOVE gives a payoff of:


(1) f*(4.5) + (1-f)*(21.5*e - 20*(1-e))


where f is the percentage of the time player 1 folds and e is the equity that player 2 has when player 1 calls. This e is obviously a function of the cards that player 2 holds, and the range of cards that player 1 is calling with. If player 1 plays RAISE/CALL every time, then f = 0, and e for a given hand for player 2 will just be the equity that hand has against a random hand (in Poker Stove, just click the 'RD' button to give a player a random range). Setting f = 0, we reduce (1) to:


(2) 21.5*e - 20*(1-e)


For SHOVE to be the correct play, it has to offer a higher payoff than the alternative, which is FOLD. Set (2) equal to the payoff for FOLD, which is zero, and solve for e :


(3) 21.5*e - 20*(1-e) = 0


=> e = 20/41.5


That is, if player 2's hand has equity of more than 20/41.5 against a random hand, he should shove. These hands are 22+,A2+,K2+,Q3o+,Q2s+,J7o+,J3s+,T8o+,T6s+, and 97s+. This is 54.8% of all hands. So how is each player doing under these strategies?


Player 1 raises every time. 45.2% of the time, player 2 folds and player 1 gets 1.5. The other 54.8% of the time, they get all in and player 1 has an average of 43% equity:

equity win tie pots won pots tied
Hand 0: 57.233% 55.68% 01.55% 847980578496 23586063110.00 { 22+, A2s+, K2s+, Q2s+, J3s+, T6s+, 97s+, 87s, A2o+, K2o+, Q3o+, J7o+, T8o+ }
Hand 1: 42.767% 41.22% 01.55% 627684857684 23586063110.00 { random }


So his total payoff is: .452*1.5 + .548*(.428*21.5 - .572*20) = -.55


Player 2 folds 45.2% of the time for a payoff of zero. The rest of the time, he has 57.2% equity for a total payoff of: .452*0 + .548*(.572*21.5 - .428*20) = 2.04 (Note that the two average payoffs add to 1.5. This must be the case under any strategies the players play, since no matter what happens, the two players' payoffs add to 1.5 - go back to the earlier post if you want to check)


Again, since we have figured out player 2's BEST response for player 1's strategy, this must be the highest payoff player 2 can get. Obviously this is not the case for player 1. If he just folded every time and got zero, he'd be better off than he would be playing the strategy of RAISE/CALL for every hand. So this is not an equilibrium (remember, an equilibrium is where both players are best responding to each other. Here, player 2 is best-responding to player but not vice versa).


Now that we've figured out player 2's best responses for some very simple player 1 strategies, let's figure it out for a strategy where player 1 varies his strategy based on his hand. Say, for instance, Player 1 plays RAISE/CALL with 22+, A8o+,A2s+,KT+, RAISE/FOLD with 45s-QJs, 68s-QTs, 89o-QJo, and T8o-QTo, and FOLD with all other hands. (1) still holds for determining player 2's payoff. To get the answer exactly right, we would have to calculate a different f for every hand player 2 has, due to card removal effects (e.g., if player 2 holds A2, it is less likely that player 1 holds AA, etc.). But we will sacrifice some accuracy in exchange for ease of calculation and calculate f independent of player 2's holdings. Notice that f is just the probability player 1 plays RAISE/FOLD divided by the sum of the probabilities player 1 plays RAISE/FOLD plus the probability he plays RAISE/CALL. That is, it's the probability he folds given that he has raised. 22+,A8o+,A2s+,KT+ is 18.6% of all hands. 45s-QJs, 68s-QTs, 89o-QJo,T8o-QTo is 10.2% of all hands, so


f = 10.2/(18.6 + 10.2) = 35.4


Plug this into (1), set equal to 0, and solve for e as earlier. you get e = .413. That is, any hand with at least 41.3% equity vs. player 1's calling range of 22+,A8o+,A2s+,KT+, player 2 should shove. Here are all such hands: 22+,A9o+,A5s+,KQo,KJs+. So player 2 is shoving 14.6% of the time. When player 1 calls, he has an average equity of 59.7%.


Now player 1's payoff is: .712*0 + .102*.146*(-3) + .186*.146*(.597*21.5 - .403*20) + .288*(1-.146)*1.5 = .45


So at least player 1 has improved upon his RAISE/CALL strategy and has now found a strategy that's better than folding every time.


Player 2's payoff is: .712*1.5 + .288*(.01*0 + .99*(.646*4.5 + .354*(.36*21.5 - .64*20)) = 1.05


So now we have figured out how to determine player 2's best response to a given strategy for player 1. Step 1: Calculate f from player 1's strategy. Step 2: Set equation (1) equal to zero and solve for e. Step 3: Go to Poker Stove and find all the hands that have at least e equity vs. player 1's calling range. This mapping from player 1's strategy to player 2's best response is called player 2's "best response function". In the next post on The Shortstack Game, we will figure out player 1's best response function. Then we will find a point where they intersect, which will then be an equilibrium. If you want to get a head start, try and find player 1's best response to player 2 playing FOLD every time, playing SHOVE every time, and playing SHOVE with the 22+,A9o+,A5s+,KQo,KJs+.


-BRUECHIPS

Brack Is Beautifur (Part 10)

For those of you looking forward to further analysis of The Shortstack Game, don't worry, that will come tomorrow. But I just had to post this first, especially because of the side action I had with Brackchips, who was watching me pray this hand. This was down to two tables of a DNG that started with 5 tables.

Seat 1: MP (600)
Seat 7: bruechips (BB) (5,860)
Seat 8: UTG (13,765)

*** HOLE CARDS ***
Dealt to bruechips [Qs Js]
UTG raises to 480
MP raises to 600, and is all in
bruechips calls 360 (This call is a little bit loose, but I did it for two reasons: 1) UTG was a megadonk, and I wanted to play hands with him, even OOP, and 2) MP was a shorty, and his less-than-min re-raise all-in meant that UTG couldn't re-raise, and I might get to check down and see the whole board for free.)
UTG calls 120

*** FLOP *** [5s Ks As] (Spades are gold)
bruechips checks
UTG bets 1,920
bruechips calls 1,920 (Looking back, maybe I should be raising this instead since UTG was a total station and his bet into the dry side pot indicates he has at least a pair and therefore would probably call if I shoved, whereas he might get scared if another spade rolls off on the turn.)

*** TURN *** [5s Ks As] [7s]
bruechips checks (Another spade...kind of kills my action...maybe he has like 8c8s and will shove all-in, but betting here would be bad as he might be able to fold some pairs.)
UTG checks

*** RIVER *** [5s Ks As 7s] [8s]
bruechips bets 3,340, and is all in (Here is the main interesting decision in the hand. The question is, is he more likely to call a shove or shove if I check? If he has a spade he MIGHT check behind but will definitely be calling if I shove. Actually this guy is such a donk, I think he's calling a shove with any two cards, thinking I'm bluffing and hoping for a split. Even though this is a bet into a side pot and I'm going to have to show down for the main anyway. So I shoved and he called. Before looking back into the 'Last Hand' tab to see what he had, I bet Brackchips $1 that he did not have a spade.)
UTG calls 3,340

*** SHOW DOWN ***
bruechips shows [Qs Js] a flush, Ace high
UTG mucked [Tc Ac] - a flush, Ace high (PWND!!! Ship the pot AND the $1!!!)

-BRUECHIPS

The Shortstack Game, Part 1

A recent post by gnome got me thinking, in part because I think he made an error in his post, which I describe in my comment there. As you might know if you follow the blog closely, I am a grad student in Economics. One of the things I study is game theory. When game theorists "solve" a game, what they look for (or at least the first thing they look for) is a "Nash equilibrium". When two players's strategies are in Nash equilibrium, each one knows the other's strategy, and even with that knowledge, neither one would choose a different strategy. Here's another way of saying it. Take some game with two players. Fix a strategy for player 2. Call player 1's "best response" to that strategy as the strategy available to him that maximally exploits player 2's strategy. Two strategies strategy1 (player 1's strategy) and strategy2 (player 2's strategy) are in Nash equilibrium if and only if strategy1 is player 1's best response to strategy2, and strategy2 is player 2's best response to strategy1.

As a whole, poker is far too complicated a game to find equilibrium strategies. There is not any guarantee that there exists a unique equilibrium. But sometimes we can take toy games that mimic at least some situations in poker, with some simplifications, and find an equilibrium. You'll find many such games in The Mathematics of Poker by Ankenmann and Chen, which I highly recommend. In any case, here we will solve 'The Shortstack Game', which works as follows:

Player 1 and Player 2 receive two cards. Player 1 can either fold or raise. If player 1 folds, he gets a payoff of zero and player 2 gets a payoff of 1.5. If player 1 raises, player 2 can either fold or shove. If player 2 folds, player 1 gets a payoff of 1.5 and player 2 gets a payoff of zero. If player 2 shoves, player 1 can either call or fold. If player 1 folds, he gets a payoff of -3 and player 2 gets a payoff of 4.5. If player 1 calls, a Hold 'Em board is dealt out and the player with the best hand wins. The winner gets a payoff of 21.5, whereas the loser gets a payoff of -20.

This game preserves the important features of a Hold 'Em situation where a player raises on the cutoff with a shorty on the button. I have made some simplifying assumptions, some of them important, some not. For instance, I say that when player 1 folds, player 2 also gets a payoff of 1.5. In reality, of course, player 2 won't always win the blinds when player 1 folds, but insteads enters into a new game with the players in the blinds. But that is inconsequential to our analysis here (although I plan to come back to it later), as all the decisions we're interested in are 1) whether player 1 decides to raise, and 2) what happens after player 1 raises. Neither of these depend at all on what player 2's payoff is when player 1 folds (convince yourself this is true or ask for clarification in comments if it is unclear).

Another assumption I've made is that the blinds fold every time (if player 1 raises and player 2 folds, player 1 wins 1.5, the blinds, automatically, and if player 2 shoves, the action is immediately back on player 1). This is obviously consequential as the potential for blinds calling or re-raising affects the payoff of player 1 when he raises and player 2 folds, and therefore how good raising is relative to folding for player 1. I could eliminate this need for simplification by making player 1 the small blind and player 2 the big blind, but: 1) If I recall correctly, there is already analysis of some very similar game in Ankenmann and Chen, and 2) Often as a good deep-stacked player you're going to want to be raising in position vs. the blinds so you can take a flop in position. I'd venture that 1.5 (i.e., winning the blinds for sure) is actually much lower than player 1's true EV of player 2 folding on the button to his raise.

Anyway, this post is plenty long enough already, so I think I'll stop there for now and allow for digestion and questions of the game setup before proceeding to solving the game. But if you want to do some work on it before next post, note that a strategy for player 1 consists of choosing either FOLD, RAISE/CALL, or RAISE/FOLD for every hand possible. A strategy for player 2 consists of choosing FOLD or SHOVE for every hand possible. What is player 2's best response to player 1 choosing RAISE/FOLD for every hand? What is player 2's best response to player 1 choosing RAISE/CALL for every hand?

-BRUECHIPS